Math 327
Matrix Multiplication Handout
Definitions:
•Given two n-dimensional vectors in Rn,~a =
a1
a2
.
.
.
an
and ~
b=
b1
b2
.
.
.
bn
, the dot product (or inner pro duct) of
~a and ~
b,~a ·~
b=a1b1+a2b2+···+anbn=
n
X
i=1
aibi
Notice that the dot product of two vectors is always a scalar.
Example: Let ~a =
−1
3
4
and ~
b=
2
−1
0
.
Then ~a ·~
b= (−1)(2) + (3)(−1) + (4)(0) = −2−3 + 0 = −5.
•Suppose that A= [aij ] is an m×pmatrix and B= [bij ] is a p×nmatrix. Then the product of Aand B, written AB
is the m×nmatrix C= [cij] defined by: cij =ai1b1j+ai2b2j+···aipbpj =
p
X
k=1
aikbkj .
Note: In order to multiply two matrices, the matrices must be of appropriate sizes relative to one another. More
specifically, the number of columns in the first matrix must be that same as the number of rows in the second matrix.
This definition may seem a bit strange, but here is an illustration of how it works:
a11 a12 ··· a1p
.
.
..
.
.··· .
.
.
ai1ai2··· aip
.
.
..
.
.··· .
.
.
am1am2··· amp
b11 b12 ··· b1j··· b1n
b21 b22 ··· b2j··· b2n
.
.
..
.
..
.
..
.
.
.
.
..
.
..
.
..
.
.
bp1bp2··· bpj ··· bpn
=
c11 c12 ··· c1n
.
.
..
.
.
cij
.
.
..
.
.
cm1ac2··· cmn
Notice that cij = [rowi(A)]T·[columnj(B)] =
p
X
k=i
aikbkj .
Example 1: Let
A=
2−1 3
4 0 5
−162
B=
4 0
−2 5
6−1
.
Find AB (Notice that the (2,1) entry of AB is found by computing [row2(A)]T·[column1(B)])
Notice that BA is undefined.
